[jjl3rd]

From an electrical point of view what has happened is that your switch is being shunted by an extra connection in parrallel with the switch. There are two likely locations where this could occur. The first place to look is at the switch itself, which you have already done.
Since its not that and you have indicated that the dishwasher and the disposal are connected to the same outlet you need to look at the outlet itself. When your system was originally wired it probably had power going directly to the outlet that the dish washer is plugged into, then the little shorting bar on the black side of the duplex outlet was removed and the switch was wired across in place of the removed bar. It sounds like the outlet which has partially melted has deformed enough to allow the black side connections to touch either inside the outlet or on its surface. You need to replace the outlet. Don't forget to remove the little shorting bar line (black) side only.
While your in there try to find out the root cause of the problem, why did the outlet melt in the first place. Is it becoming wet? Have caustic cleaning agents under the sink dammaged the outlet? IS the wiring aluminum?